U.S. patent application number 10/340847 was filed with the patent office on 2003-08-28 for neuronal network for modeling a physical system, and a method for forming such a neuronal network.
Invention is credited to Seifert, Jost.
Application Number | 20030163436 10/340847 |
Document ID | / |
Family ID | 7712023 |
Filed Date | 2003-08-28 |
United States Patent
Application |
20030163436 |
Kind Code |
A1 |
Seifert, Jost |
August 28, 2003 |
Neuronal network for modeling a physical system, and a method for
forming such a neuronal network
Abstract
A neuronal network for modeling an output function that
describes a physical system using functionally linked neurons (2),
each of which is assigned a transfer function, allowing it to
transfer an output value determined from said neuron to the next
neuron that is functionally connected to it in series in the
longitudinal direction (6) of the network (1), as an input value.
The functional relations necessary for linking the neurons are
provided within only one of at least two groups (21, 22, 23) of
neurons arranged in a transverse direction (7) and between one
input layer (3) and one output layer (5). The groups (21, 22, 23)
include at least two intermediate layers (11, 12, 13) arranged
sequentially in a longitudinal direction (5), each with at least
one neuron.
Inventors: |
Seifert, Jost; (Schliersee,
DE) |
Correspondence
Address: |
CROWELL & MORING LLP
INTELLECTUAL PROPERTY GROUP
P.O. BOX 14300
WASHINGTON
DC
20044-4300
US
|
Family ID: |
7712023 |
Appl. No.: |
10/340847 |
Filed: |
January 13, 2003 |
Current U.S.
Class: |
706/26 ;
706/31 |
Current CPC
Class: |
G06N 3/0454
20130101 |
Class at
Publication: |
706/26 ;
706/31 |
International
Class: |
G06N 003/067; G06N
003/063; G06N 003/06; G06F 015/18; G06G 007/00; G06E 003/00; G06E
001/00 |
Foreign Application Data
Date |
Code |
Application Number |
Jan 11, 2002 |
DE |
102 01 018.8 |
Claims
What is claimed is:
1. A neuronal network for use in modeling a physical system
mathematically defined by a functional equation having summation
terms having subfunctions and subfunction coefficients, said
network comprising: an input layer; an output layer; and a group
layer, said group layer including at least two groups of neurons,
wherein the number of groups of neurons is equal to the number of
subfunctions in the functional equation being used to describe the
system being modeled, and wherein the subfunction coefficients are
arranged in the form of untrainable input links, after an output
neuron in a respective group.
2. The neuronal network in accordance with claim 1, wherein the
input and output layers include respective input and output neurons
which are linear, in order to allow a transfer of input values,
unchanged, to the neurons in a first layer of a group, and in order
to avoid limiting the value range for the output of the neuronal
network.
3. The neuronal network in accordance with claim 1, further
including fixed value untrainable links between the neurons in the
input layer and the first layer in a group, wherein said fixed
value is determined during the training of the neuronal
network.
4. The neuronal network in accordance with claim 1, further
including an untrainable input link for the multiplication of the
output of a group with a predetermined factor.
5. The neuronal network in accordance with claim 1, wherein the
neuronal network is used to set up a simulation model.
6. The neuronal network in accordance with claim 1, wherein the
neuronal network is analyzed by viewing one group as an isolated,
neuronal network, wherein a first intermediate layer becomes the
input layer and a last intermediate layer becomes the output
layer.
7. The neuronal network in accordance with claim 1, wherein one
group is trained in isolation, in which only link weights of a
group are modified, using a training data set and an optimization
process.
8. The neuronal network in accordance with claim 1, wherein a value
range of a group is defined via a suitable selection of a transfer
function for the output neuron of a group.
9. A neuronal network for use in modeling an output function that
describes a physical system, said network comprising functionally
connected neurons, each of which is assigned a transfer function,
allowing transfer of a determined output value as an input value to
a next neuron functionally connected in series, in the longitudinal
direction of the network, wherein functional relations for linking
the neurons are provided within only one of at least two groups of
neurons, arranged in a transverse direction between an input layer
and an output layer, and wherein each of the at least two the
groups of neurons comprise at least two intermediate layers,
arranged sequentially in a longitudinal direction, each of said at
least two intermediate layers having at least one neuron, wherein
the subfunction coefficients are considered in the form of
untrainable links between a neuron group and the output layer
neurons in the entire neuronal network, and are provided as links
between the input layer and each one of a group of untrainable
input links.
10. The neuronal network for modeling an output function that
describes a physical system in accordance with claim 9, wherein a
number of neuronal groups is equal to a number of subfunctions in a
functional equation that describes the system being simulated.
11. The neuronal network for modeling an output function that
describes a physical system according to claim 9, wherein the input
layer of neurons and the output layer of neurons of the neuronal
network are linear.
12. A method for setting up a neuronal network in accordance with
claim 1, comprising: adopting the values for the input into the
network from a training data set; registering the input neurons and
the input links with said adopted values; calculating the neuronal
network from the input layer up to the output layer, wherein an
activation of each neuron is calculated dependent upon preceding
neurons and links; comparing said neurons with a reference value
from the training data set, and calculating the network error from
the difference in order to activate the output neurons, wherein the
error for each neuron is calculated from the network error, in
layers from the back to the front; calculating the weight change in
links to adjacent neurons, dependent upon the error of one neuron
and its activation, wherein one of untrainable input links and the
untrainable links are excluded; and adding the calculated weight
changes to proper link weights, wherein the untrainable input links
and the untrainable links are excluded.
13. The method for training a neuronal network in accordance with
claim 9 for implementation in a computer program system, wherein
the input and output values of the system are measured and a
training data set for the neuronal network is established from the
measured data, with these data being formed from a number of value
pairs, each comprising four input values (.alpha., M.alpha., .eta.,
q) and one output value (C.sub.M).
14. The method of training, in accordance with claim 13, wherein
the method of descent by degree is used.
15. The optimization process for adjusting the link weights of a
neuronal network in accordance with claim 9 for implementation in a
computer program system, wherein trainable link weights w.sub.y are
adjusted such that the neuronal network supplies an optimal output
for all measured data, wherein the values for the inputs to the
network are taken from the training data set, comprising: assigning
random values to the link weights; setting the input links to the
input values from the training data set; calculating the network
from the input layer up to the output layer, wherein the activation
of each neuron is calculated independent of the preceding neurons
and links; comparing the activation of the output neurons with the
reference value from the training data set, and calculating the
network error from the difference; calculating for each layer of
the error at each neuron from the network error, against the
longitudinal orientation, wherein the links function as inputs;
calculating the weight change in the links to adjacent neurons,
dependent upon the error of one neuron and its activation; adding
the weight changes to the proper link weights, wherein the weight
changes are not added to the untrainable links and the untrainable
input links.
16. The optimization process for adjusting the link weights of a
neuronal network in accordance with claim 15, wherein the link
weights are set to random values within the range of [-1.0 to
+1.0].
17. The optimization process for adjusting the link weights of a
neuronal network in accordance with claim 15, wherein the
optimization process is conducted for only one group in the
neuronal network.
Description
BACKGROUND AND SUMMARY OF THE INVENTION
[0001] This application claims the priority of Application No. 102
01 018.8, filed Jan. 11, 2002, in Germany, the disclosure of which
is expressly incorporated by reference herein.
[0002] The invention relates to a neuronal network for modeling a
physical system using a computer program system for system
identification, and a method for forming such a neuronal network,
wherein the invention can be used for physical systems that are
dynamically variable.
[0003] Systems that are suitable for application with this network
are those that fall within the realm of movable objects such as
vehicles, especially aircraft, and systems involving dynamic
processes such as reactors and power plants, or chemical processes.
The invention is especially well suited for use in modeling
vehicles, especially aircraft, using aerodynamic coefficients.
[0004] In a system identification process for the formation of
analytical models of a physical system, it is important to
reproduce the performance characteristics of the system with its
inputs and outputs as precisely as possible, in order that it may
be used, for example, in simulations and for further testing of the
physical system. The analytical model is a mathematical model of
the physical system to be copied and should produce output values
that are as close as possible to those of the real system, with the
same input values. The following are ordinarily required for the
modeling of a physical system:
[0005] Pairs of measured input and output values
[0006] A model structure
[0007] A method for determining characteristic values
[0008] In some processes, estimated initial values for the
characteristic values.
[0009] To simulate aircraft using aerodynamic coefficients, a
determination of aerodynamic coefficients is necessary, which, in
the current state of the art, is accomplished via the so-called
"Equation Error Method" and the so-called "Output Error
Method".
[0010] In these methods, the performance characteristics of the
system are simulated using linear correlations, wherein a precise
understanding of the model and an undisrupted measurement are
ordinarily assumed. These methods carry with them the following
disadvantages:
[0011] a) Ordinarily, a linear performance characteristic
describing an initial state is required. Consequently, it is
difficult to reproduce a highly dynamic performance characteristic
correctly for a system, since state-dependent characteristic values
are no longer in linear correlation with the initial state.
[0012] b) Relevant characteristic values can be identified only for
particular portions of the measured values (e.g., aircraft
maneuvers). This results in high data processing costs.
[0013] c) A convergence of the methods can be impeded by
sensitivity to outdated measured data.
[0014] As an alternative to these established methods, neuronal
networks are used in system modeling. Due to the relatively high
level of networking of the neurons, multi-layered, forward-directed
networks are used which are similar to a black-box, whereby a
characteristic value of the modeled system cannot be localized.
This means that internal dimensions of the network cannot be
assigned specific physical effects; hence, they cannot be analyzed
in detail. This type of analysis is important, however, for the
formulation of statements regarding the general effectiveness of
the overall model. Due to this black-box character, neuronal
networks have thus far not been used for system identification.
[0015] It is the object of the invention to create a neuronal
network for modeling a physical system using a computer program
system for system identification, and a method for constructing
said network. The network must be robust and permit the
determination of characteristic values for the modeled system.
[0016] According to the invention, a neuronal network is provided
for modeling an output function that describes a physical system,
consisting of neurons that are functionally connected to one
another. A transfer function is assigned to each of the neurons,
allowing them to transfer the output value determined from that
neuron to the neuron that is functionally connected to it in
sequence, in the longitudinal direction of the network, as an input
value. The functional relations for connecting the neurons are
provided within only one of at least two groups of neurons that are
arranged in a transverse direction between an input layer and an
output layer, wherein the groups include at least two intermediate
layers arranged sequentially in a longitudinal direction and have
at least one neuron each. In particular, the subfunction
coefficients in the form of untrainable links between a group of
neurons and the output neurons of the entire neuronal network are
considered. They are provided as links between the input layer and
each group of untrainable input links.
[0017] With the structure of the neuronal network provided in the
invention it is possible to assign specific physical effects to
individual neurons, which is not possible with the current state of
the art neuronal networks that lack the system-describing model
structure. In general, the neuronal network specified in the
invention ensures greater robustness than outdated measured data,
and furthermore offers the advantage over the "Equation Error
Method" and the "Output Error Method" that functions to describing
the system to be modeled, allowing an improved manipulation of the
invention when used on similar systems.
[0018] According to the invention, a neuronal network is provided
for use in the formation of analytical models of physical systems,
wherein the dynamic and physical correlations of the system can be
modeled in a network structure. To this end, it is necessary that
the output of the system be comprised of a sum of a number of parts
(at least two), which are calculated from the input values. For
each part, a physical effect (e.g., the stabilization of a system)
can be defined.
[0019] The method specified in the invention offers the following
advantages: With the use of neuronal networks as described in the
invention, a greater robustness is achieved over outdated measured
data, and the analytical model is not limited to linear
correlations in the system description, since output values for all
input values within a preset value range are interpolated or
extrapolated in a non-linear manner. Furthermore, with the use of
the neuronal network specified in the invention, a generalization
can be made, i.e., general overall trends can be derived from
erroneous measured data.
[0020] Furthermore, due to the structure of the neuronal network
specified in the invention, specific expert knowledge regarding the
modeled physical system can also be incorporated via a specific
network structure and predefined value ranges.
[0021] Other objects, advantages and novel features of the present
invention will become apparent from the following detailed
description of the invention when considered in conjunction with
the accompany drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
[0022] The invention will be described with reference to the
attached figures, which show:
[0023] FIG. 1 is an exemplary embodiment of a neuronal network
being used to form an analytical model of a physical system being
reproduced, as specified in the invention,
[0024] FIG. 2 is a representation of a neuronal network according
to the general state of the art.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
[0025] The neuronal network specified in the invention for use in
modeling an output function that describes a physical system is
comprised of functionally connected neurons (2), each of which is
assigned a transfer function, allowing it to transfer the output
value determined from that neuron, as an input value, to the neuron
2 that in the longitudinal direction 6 of the network 1 is
functionally connected to it as the next neuron. In the following
description, terms ordinarily associated with neuronal networks
such as layers, neurons, and links between the neurons will be
used. Additionally, the following nomenclature will be used:
[0026] O.sub.l Output of a neuron from the preceding layer l,
[0027] W.sub.lj Trainable link weight between two layers l and
j,
[0028] .function..sub.j Transfer function of a neuron in the
subsequent layer j.
1 W.sub.lj W.sub.lj O.sub.1 .fwdarw. .DELTA. .function..sub.j =
tanh(.SIGMA.o.sub.lw.sub.l]) O.sub.1 .fwdarw. .O slashed.
.function..sub.j = .SIGMA.o.sub.jw.sub.lj Neuron with non-linear
transfer function Neuron with linear transfer function
[0029] The neuronal network specified in the invention is based
upon analytical equations for describing the performance
characteristics of the system, dependent upon input values. These
equations comprise factors and functions of varying dimensions.
These functions can be linear or non-linear. To describe the system
in accordance with the method specified in the invention using a
neuronal network, these functions and their parameters must be
established, wherein neurons with non-linear or linear transfer
functions are used.
[0030] One exemplary embodiment of a neuronal network, as specified
in the invention, is represented in FIG. 1 for an aerodynamic model
describing the longitudinal movement of an aircraft. According to
the invention, multilayer feed-forward networks (Multi Layer
Perception) are used. With the network structure specified in the
invention, and with the modified optimization process, a separation
of the physical effects and an assignment of these effects to
prepared groups take place. Each group represents a physical effect
and can, following a successful training of the entire network, be
analyzed in isolation. This is because a group can also be isolated
from the overall network, and, since both inputs and outputs can be
provided for any input values, output values for the group can also
be calculated.
[0031] A neuronal network according to the current state of the
art, with neurons having a non-linear transfer function for the
construction of a function .function. having four input values x,
y, a, b is represented in FIG. 2. The illustrated neuronal network
100 is provided with an input layer 101 with input neurons 101a,
101b, 101x, 101y, an output neuron 104, and a first 111 and a
second 112 intermediate layer. The number of intermediate layers
and neurons that are ordinarily used is based upon pragmatic values
and is dependent upon the complexity of the system to be simulated.
In the traditional approach, the neurons are linked to one another
either completely or in layers. Typically, the input neurons are on
the left side, and at least one output neuron is on the right side.
Neurons can generally have a non-linear function, e.g., formed via
the tangent hyperbolic function, or a linear transfer function. The
neurons used in these figures are hereinafter referred to using the
corresponding reference symbols. Due to its cross-linked structure,
these parts cannot determine or solve the system equation for the
parameters.
[0032] In accordance with the invention, to solve an equation to
describe a physical system, a neuronal network having a specific
architecture is used (see FIG. 1). While intermediate layers
arranged sequentially as viewed in the longitudinal direction 6 of
the network 1, which hereinafter are referred to in combination as
a group layer 4, are retained, at least two additional groups of
neurons are formed, arranged in a transverse direction 7. In
contrast to the traditional arrangement, the formation of groups
allows the partial subfunctions to be considered individually.
[0033] According to the invention, the functional relations for
connecting the neurons are provided within only one of at least two
groups 21, 22, 23 of neurons, arranged in a transverse direction 7
and between an input layer 3 and an output layer 5, wherein the
groups 21, 22, 23 comprise at least two intermediate layers 11, 12,
13 arranged sequentially in a longitudinal direction 5, and
comprising at least one neuron. Thus one neuron in an intermediate
layer is connected to only one neuron in another, adjacent
intermediate layer, via functional relations that extend in the
longitudinal direction 6 of the network 1, with these neurons
belonging to one of several groups of at least one neuron each,
arranged in a transverse direction 7. The groups of neurons are
thus isolated, i.e., the neurons of one group of neurons are not
directly connected to the neurons of another group. Within a group
of neurons, any number of intermediate layers may be contained.
[0034] The groups of neurons used in the invention comprise at
least one input layer 3 having at least one input neuron (reference
figures x and y; the references x and y are also used for the
corresponding variables or input values), and at least one output
layer 5 having at least one output neuron 9.
[0035] The number of neuron groups to be formed in accordance with
the invention is preferably equal to the number of subfunctions in
the functional equation being used to describe the system being
simulated.
[0036] Advantageously, in the architecture specified in the
invention, the subfunction coefficients are integrated in the form
of untrainable input links behind the group layer. In this way, the
number of links, and thus also the time required for training and
calculating, is reduced. In state-of-the-art neuronal networks, in
contrast, these subfunction coefficients would be in the form of
input neurons (FIG. 2).
[0037] The input and output neurons in the neuronal network are
preferably linear, in order to pass on the input values, unchanged,
to the groups, and in order to simply add up the outputs from the
groups.
[0038] A group of neurons is connected to the input neurons via
untrainable links, and to the output neurons of the entire neuronal
network via untrainable input links.
[0039] With the untrainable input link, the output of a group of
neurons can still be multiplied by a factor (e.g., 12(x, y)
multiplied by a).
[0040] The untrainable input links are advantageously used to
assign physical effects to prepared groups. These links enable the
calculated total error at the network output to be split up into
the individual parts from the groups, during the optimization
process (training). Thus, for example, with an input link having
the value of zero, this group cannot have contributed to the total
error. Hence, the value of zero is calculated as a back-propagated
error in accordance with the back-propagation algorithm. The
error-dependent adjustment of the weights within this group is thus
avoided. Only those groups whose untrainable input links are not
equal to zero are adjusted.
[0041] Below, this network architecture is described by way of
example, using a physical system having the following mathematical
approximation:
.function.(x,y,a,b)=.function.2(x,y)+.function.2(x,y).multidot.a+.function-
.3(y).multidot.b (1)
[0042] This type of function can be used to describe a multitude of
physical systems, such as the formula given in the equation (2) for
the longitudinal movement (pitch momentum) of an aircraft:
C.sub.M=C.sub.M0(a, Ma)+C.sub.M.eta.(a,
Ma).multidot..eta.+C.sub.Mq(Ma).mu- ltidot.q (2)
[0043] In the representation of the equation (1), the coefficients
are the functions .function.1, .function.2 and .function.3, and in
the representation of the equation (2) they are C.sub.M0,
C.sub.M.eta., and C.sub.Mq. These individual coefficients are
generally non-linearly dependent upon the angle of pitch a and
sometimes upon the Mach number Ma.
[0044] In this:
2 C.sub.M = pitch momentum coefficient C.sub.M0(a, Ma) = zero
momentum coefficient, dependent upon the pitch angle a and the Mach
number Ma; C.sub.M.eta.(a, Ma) = derivative for the increase in
pitch momentum resulting from elevator control deflection; it is
dependent upon the pitch angle a and the Mach number Ma, and must
be multiplied by .eta. C.sub.Mq(Ma) = derivative for stabilization
of pitch; it is dependent
[0045] upon the Mach number Ma, and must be multiplied by the pitch
rate q.
[0046] FIG. 1 shows a neuronal network 1 formed from neurons 2 and
based upon the starting equation (1), used by way of example, with
said network comprising an input layer 3 and an output layer 5, and
several, at least two, groups in the group layer 4. Within one
group, a first 11, a second 12, and a third 13 intermediate layer
are arranged--each as a component of the group layer 4. The number
of intermediate layers that are used is dependent upon the order of
the function to be approximated, with which the simulated system is
mathematically described. Ordinarily one to three intermediate
layers are used.
[0047] According to the invention, groups of neurons are formed in
the group layer, arranged in the network 1 in a transverse
direction 7, wherein the number of neuron groups to be formed in
accordance with the invention is preferably equal to the number of
subfunctions in the functional equation being used to describe the
system being simulated. In the equation (1) and/or the equation
(2), there are three subfunctions as their specialization.
Accordingly, in the embodiment shown in FIG. 1, three neuron groups
21, 22, 23 are provided. In this manner, with the formation of
groups arranged in a transverse direction, the given subfunctions,
which in the example of the equation (1) are the functions
.function.1, .function.2, and .function.3, can be viewed in
isolation. To this end, the first intermediate layer 11 is used as
an input layer and the last intermediate layer 13 is used as an
output layer.
[0048] In the neuronal network formed for the equation (1) in FIG.
1, the subfunction coefficients are the coefficients 1, a and b,
and are integrated into the overall network in the form of
untrainable input links 8b; i.e., the links between the last
intermediate layer 13 of a group and the output layer 5 are acted
upon with the functional coefficients. In this manner, the number
of links, and thus also the time required for training and
calculation, is reduced. The input and output neurons of the group,
in other words the input layer 3 and the output layer 5, should
preferably be linear, in order to allow the input values to be
passed on, unchanged, to the neurons of the intermediate layers 11,
12, 13, and to allow the output values for the neuron groups to be
simply added up.
[0049] The neuron groups 21, 22, 23 used in the invention comprise
a first intermediate layer, or input intermediate layer 11 in the
group layer 4, with at least one input neuron 31a or 32a, 32b, or
33a, 33b. A last intermediate layer or output intermediate layer 13
comprises at least one output neuron 31c or 32b or 33c. The neuron
groups 21, 22, 23, which are functionally independent of one
another due to the functional correlations in the transverse
direction 7, are isolated from one another, i.e., the neurons of
one neuron group are not directly linked to the neurons of another
neuron group. This does not apply to the functional link to the
input layer 3 and the output layer 5. Any number of intermediate
layers can be contained within a neuron group. In the exemplary
embodiment shown in FIG. 1 three intermediate layers 12 are
arranged. This means that, according to the invention, the
functional relations for linking the neurons are provided within
only one of at least two groups of neurons 21, 22, 23 that are
arranged in a transverse direction 7 and between an input layer 3
and an output layer 5. Each group 21, 22, 23 comprises at least two
intermediate layers 11, 12, 13 arranged sequentially in a
longitudinal direction 6, each with at least one neuron. Thus, one
neuron in an intermediate layer is connected to only one neuron in
another, adjacent intermediate layer, via functional relations that
extend in a longitudinal direction 6 in the network 1, when these
neurons belong to one of several groups arranged in a transverse
direction 7 and containing at least one neuron each.
[0050] With the neuronal network specified in the invention, the
internal terms .function.1, .function.2, .function.3 of the
equation (1), and/or the terms C.sub.M0(a,Ma), C.sub.M.eta.(a,Ma),
C.sub.Mq(Ma) in the more specialized equation (2) can be determined
using the network parameters (link weights), in order to test the
model for the proper performance characteristics with untrained
input values. For example, with the equation (2) the term
C.sub.Mq(Ma) should always be negative, because it represents the
stabilization of the system. These analytical possibilities are
achieved via the architecture of the neuronal network used in
accordance with the invention (see FIG. 1).
[0051] The method for adjusting or defining the neuronal network
specified in the invention will now be described in greater
detail:
[0052] To form models of dynamic systems, analytical equations
designed to describe the system's performance characteristics,
dependent upon input values, are set up. One example of such an
equation is formulated above in the equation (1). These equations
comprise factors and functions of varying dimensions. These
functions can be linear or non-linear. In a further step in the
method specified in the invention, these functions and their
parameters that describe the system being modeled must be
determined. The structure of the neuronal network is then
established according to the above-described criteria. One
exemplary embodiment of a neuronal network used in accordance with
the invention is represented in FIG. 1 for an aerodynamic model
that describes the longitudinal movement of an aircraft. The
architecture of the neuronal network 1 is structured analogous to
the mathematical function .function.(x,y,a,b), wherein untrainable
links 8a are provided between the input layer and the first group
layer 11, and untrainable input links 8b are provided between the
last group layer 13 and the output layer 5.
[0053] A training phase follows, during which the network is
adjusted to agree with the system being simulated. In this, the
input and output values for the system (in this case an aircraft)
are measured. For the aerodynamic example, the mechanical flight
values .alpha., M.alpha., .eta., q and C.sub.M are measured or
calculated using mechanical flight formulas. From the measured
data, a training data set is established for the neuronal network,
comprised of a number of value pairs, each containing four input
values (.alpha., M.alpha., .eta., q) and one output value
(C.sub.M). Iterative processes, e.g., the method of descent by
degree (back propagation), can be used in the learning process. In
this, to optimize the neuronal network, the trainable link weights
w.sub.y (indicated here as arrows) are ordinarily adjusted such
that the neuronal network will supply the best possible output for
all the measured data.
[0054] An optimization process is then implemented using the
training data set to establish the link weights for the neuronal
network. In this manner, the parts .function.1, .function.2 and
.function.3 can be represented exclusively in the groups provided
for this purpose.
[0055] Prior to optimization, all link weights can be set as random
values, preferably within the range [-1.0:+1.0]. If preset values
exist for the terms .function.1, .function.2, and .function.3, the
groups may also be individually pretrained. To accomplish this, a
group must be considered a closed neuronal network, and the
optimization algorithm must be used on this group alone.
[0056] The optimization of the link weights in accordance with the
known back-propagation algorithm is accomplished via the following
steps:
[0057] The values for the inputs into the network are adopted from
the training data set. In step 1, in addition to the neurons in the
input layer 3, the input links 8b must also be set to the input
values from the training data set.
[0058] The network is calculated starting with the input layer and
continuing to the output layer. In this, the activation of each
neuron is calculated based upon the preceding neurons and
links.
[0059] The activation of the output neurons is compared with the
reference value from the training data set. Network error is
calculated from the difference.
[0060] From the network error, the error in each neuron is
calculated, in layers starting from the back and traveling forward,
wherein the links can also function as inputs.
[0061] Dependent upon the error of one neuron and its activation, a
weight change in the links to adjacent neurons is calculated,
wherein the links can also function as inputs.
[0062] Finally, the weight changes are added to the proper link
weights, wherein the weight changes are not added to the
untrainable links 8a and the untrainable input links 8b.
[0063] Following the successful training of the neuronal network,
each group .function.1, .function.2 and .function.3 can be analyzed
in isolation. This is because each group can be viewed as a closed
neuronal network. In this, an input value y for the neuron 31a can
be selected, and then this group can be calculated up to its output
neuron 31c. The output neuron 31c of the group 21 then contains the
functional value .function.3(y).
[0064] For the aerodynamic example this means that:
[0065] The internal parts C.sub.Ma(a,Ma), C.sub.M.eta.(a,Ma), and
C.sub.Mq(Ma) can be provided to the output neurons 31c, 32c, 33c of
the three neuron groups following calculation of the neuronal
network 1.
[0066] The processes described, especially the process for training
and optimizing the neuronal network specified in the invention, are
intended especially for implementation in a computer program
system.
[0067] The foregoing disclosure has been set forth merely to
illustrate the invention and is not intended to be limiting. Since
modifications of the disclosed embodiments incorporating the spirit
and substance of the invention may occur to persons skilled in the
art, the invention should be construed to include everything within
the scope of the appended claims and equivalents thereof.
* * * * *